Two positive integers a and b can be written as a = x3y2 and b = xy3,

Question:

Two positive integers a and $b$ can be written as $a=x^{3} y^{2}$ and $b=x y^{3}$, where $x$ and $y$ are prime numbers. Find $\operatorname{HCF}(a, b)$ and $L C M(a, b)$.

 

Solution:

It is given that, $a=x^{3} y^{2}$ and $b=x y^{3}$, where $x$ and $y$ are prime numbers.

$\mathrm{LCM}(a, b)=\mathrm{LCM}\left(x^{3} y^{2}, x y^{3}\right)$

$=$ The highest of indices of $x$ and $y$

$=x^{3} y^{3}$

$\operatorname{HCF}(a, b)=\operatorname{HCF}\left(x^{3} y^{2}, x y^{3}\right)$

$=$ The lowest of indices of $x$ and $y$

$=x y^{2}$

Hence, $\operatorname{HCF}(a, b)=x y^{2}$ and $\operatorname{LCM}(a, b)=x^{3} y^{3}$.

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